\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small \emph{Electronic Journal of
Differential Equations}, Vol. 2010(2010), No. 68, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/68\hfil Existence of solutions]
{Existence of solutions for nonlinear parabolic systems via weak
convergence of truncations}
\author[E. Azroul, H. Redwane, M. Rhoudaf\hfil EJDE-2010/68\hfilneg]
{Elhoussine Azroul, Hicham Redwane, Mohamed Rhoudaf} % in alphabetical order
\address{Elhoussine Azroul \newline
D\'epartement de Math\'ematiques et Informatique,
Facult\'e des Sciences Dhar-Mahraz. B.P. 1796 Atlas F\`es, Morocco}
\email{azroul\_elhoussine@yahoo.fr}
\address{Hicham Redwane \newline
Facult\'e des Sciences Juridiques, Economiques et Sociales,
Universit\'e Hassan 1, B.P. 784, Settat 26 000, Morocco}
\email{redwane\_hicham@yahoo.fr}
\address{Mohamed Rhoudaf \newline
D\'epartement des Math\'ematiques,
Facult\'e des Sciences et Techniques de Tanger. B.P. 416, Tanger,
Morocco}
\email{rhoudaf\_mohamed@yahoo.fr}
\thanks{Submitted March 2, 2010. Published May 17, 2010.}
\subjclass[2000]{47A15, 46A32, 47D20}
\keywords{Nonlinear parabolic systems; existence; truncations;
\hfill\break\indent renormalized solutions}
\begin{abstract}
We prove an existence result for a class of nonlinear parabolic
systems. Without assumptions on the growth of some nonlinear
terms, we prove the existence of a renormalized solution.
\end{abstract}
\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}\label{s1}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$, ($N\geq
1$), $T>0$ and let $Q:=(0,T)\times\Omega$. We prove the existence of
a renormalized solution for the nonlinear parabolic systems
\begin{gather}\label{1}
(b_i(u_i))_t- \mathop{\rm div}\Big(a(x,u_i,
Du_i)+\Phi_i(u_i)\Big)+ f_i(x,u_1,u_2)= 0 \quad\text{in } Q, \\
\label{2}
u_i=0 \quad \text{on } \Gamma:=(0,T)\times\partial\Omega , \\
\label{3} b_i(u_i)(t=0)=b_i(u_{i,0})\quad \text{in } \Omega,
\end{gather}
where $i=1,2$. Here, the vector field
\begin{equation} \label{4}
a : \Omega\times\mathbb{R}\times\mathbb{R}^N \to \mathbb{R}^N \text{
is a Carath\'eodory function such that}
\end{equation}
\begin{itemize}
\item There exists $\alpha>0$ with
\begin{equation} \label{5}
a(x,s,\xi).\xi \geq \alpha |\xi|^p
\end{equation}
for almost every $x\in \Omega$, for every $s \in \mathbb{R}$, for
every $\xi \in \mathbb{R}^N$.
\item For each $K>0$, there exists $\beta_K>0$ and a function
$C_K$ in $L^{p'}(\Omega)$ such that
\begin{equation} \label{6}
|a(x,s,\xi)|\leq C_K(x)+\beta_K |\xi|^{p-1}
\end{equation}
for almost every $x\in \Omega$, for every $s$ such that $|s|\leq K$,
and for every $\xi\in \mathbb{R}^N$.
\item The vector field $a$ is monotone in $\xi$; i.e.,
\begin{equation} \label{7}
[ a(x,s,\xi)-a(x,s,\xi')][\xi-\xi'] \geq 0,
\end{equation}
for any $s \in \mathbb{R}$, for any $(\xi,\xi')\in \mathbb{R}^{2N}$
and for almost every $x\in \Omega$.
\end{itemize}
Moreover, we suppose that for $i=1, 2$,
\begin{gather}\label{8}
\Phi_i:\mathbb{R} \to \mathbb{R}^N \text{ is a continuous function,}\\
\label{9} b_i: \mathbb{R}\to \mathbb{R} \text{ is a strictly
increasing $C^1$-function with } b_i(0)=0,
\end{gather}
$f_i: \Omega\times\mathbb{R}\times\mathbb{R} \to \mathbb{R}$
is a Carath\'eodory function with
\begin{equation}
f_1(x,0,s)=f_2(x,s,0)=0 \quad \text{a.e. } x\in \Omega, \forall s\in
\mathbb{R}. \label{10}
\end{equation}
and for almost every $x\in \Omega$, for every $s_1, s_2\in
\mathbb{R}$,
\begin{equation}\label{11}
sign(s_i)f_i(x,s_1,s_2)\geq 0.
\end{equation}
The growth assumptions on $f_i$ are as follows: For each $K>0$,
there exists $\sigma_{K}>0$ and a function $F_{K}$ in
$L^{1}(\Omega)$ such that
\begin{equation} \label{12}
|f_1(x,s_1,s_2)|\leq F_{K}(x)+\sigma_{K}\ |b_2(s_2)|
\end{equation}
a.e. in $\Omega$, for all $s_1$ such that $|s_1|\leq K$, for all
$s_2\in \mathbb{R}$.
For each $K>0$, there exists $\lambda_K>0$ and a function $G_K$ in
$L^{1}(\Omega)$ such that
\begin{equation} \label{13}
|f_2(x,s_1,s_2)|\leq G_K(x)+\lambda_K\ |b_1(s_1)|
\end{equation}
for almost every $x\in \Omega$, for every $s_2$ such that $|s_2|\leq
K$, and for every $s_1\in \mathbb{R}$. Finally, we assume the
following condition on the initial data $u_{i,0}$:
\begin{equation}\label{14}
u_{i,0}\text{ is a measurable function such that } b_i(u_{i,0})\in
L^1(\Omega), \text{ for } i=1,2.
\end{equation}
The main difficulty when dealing with problem \eqref{1}-\eqref{3}
is due to the fact that the functions $a(x,u_i,Du_i), \Phi_i(u_i)$
and $f_i(x,u_1,u_2)$ are not in $(L^1_{loc}(Q))^N$
in general, since
the growth of $a(x,u_i,Du_i), \Phi_i(u_i)$ and $f_i(x,u_1,u_2)$ are
not controlled with respect to $u_i$, so that proving existence of a
weak solution (i.e. in the distribution meaning) seems to be an
arduous task. To overcome this difficulty, we use in this paper the
framework of renormalized solutions due to Lions and DiPerna
\cite{DL} for the study of Boltzmann equations (see also Lions
\cite{PL} for a few applications to fluid mechanics models). This
notion was then adapted to the elliptic version of
\eqref{1}-\eqref{3} in Boccardo, Diaz, Giachetti, Murat
\cite{BGDM}, in Lions and Murat \cite{M} and Murat\cite{M, FM}. At
the same the equivalent notion of entropy solutions have been
developed independently by B\'enilan and al. \cite{BBGGPV} for the
study of nonlinear elliptic problems.
The particular case where $b_i(u_i)=u_i$ and $\Phi_i=\Phi$, $i=1,2$
has been studied in Redwane \cite{R2} and for the parabolic version
of \eqref{1}-\eqref{3}, existence and uniqueness results are
already proved in \cite{BMR} (see also \cite{P} and \cite{R1}) in
the case where $f_i(x,u_1,u_2)$ is replaced by $f+ \mathop{\rm
div}(g)$ where $f\in L^1(Q)$ and $g \in {L^{p'}(Q)}^N$.
In the case where $a(t,x,s,\xi)$ is independent of $s$, $\Phi_i=0$
and $g=0$, existence and uniqueness are established in \cite{B}; in
\cite{BM}, and in the case where $a(t,x,s,\xi)$ is independent of
$s$ and linear with respect to $\xi$, existence and uniqueness are
established in \cite{BR}.
In the case where $\Phi_i=0$ and the operator
$\Delta_pu=\mathop{\rm div}|\nabla u|^{p-2}\nabla u)$ p-Laplacian
replaces a nonlinear term $\mathop{\rm div}a(x,s,\xi))$, existence
of a solution for nonlinear parabolic systems \eqref{1}-\eqref{3} is
investigated in \cite{H1, H2}, in \cite{M1} and in
\cite{E1}, where an existence result of as (usual) weak solution is proved.
This article is organized as follows: in Section \ref{s2}, we
specify the notation and give the definition of a renormalized
solution of \eqref{1}-\eqref{3}. Then, in Section \ref{s3}, we
establish the existence of such a solution (see Theorem \ref{t1}).
\section{Notation}\label{s2}
In this paper, for $K>0$, we denote by $T_K:r\mapsto
\min(K,max(r,-K))$ the truncation function at height $K$. For any
measurable subset $E$ of $Q$, we denote by $meas(E)$ the Lebesgue
measure of $E$. For any measurable function $v$ defined on $Q$ and
for any real number $s, \chi_{\{vs\}}$) denote the characteristic
function of the set $\{(x,t)\in Q\ ;\ v(x,t)s\}$).
\begin{definition}\label{d1}
A couple of functions $(u_1,u_2)$ defined on $Q$ is called a
renormalized solution of \eqref{1}-\eqref{3} if for $i=1,2$ the
function $u_i$ satisfies
\begin{equation}\label{15}
T_K(u_i) \in L^p(0,T;W^{1,p}_0(\Omega)) \quad \text{and}\quad
b_i(u_i) \in L^{\infty}(0,T;L^1(\Omega)),
\end{equation}
for any $K\geq 0$.
\begin{equation}\label{16}
\int_{\{(t,x)\in Q\ ; \ n\leq |u_i(x,t)|\leq n+1\}}a(x,u_i,Du_i)Du_i
\,dx\,dt \to 0\quad \text{as }
n \to +\infty,
\end{equation}
and if, for every function $S$ in $W^{2,\infty}(\mathbb{R})$ which
is piecewise $C^1$ and such that $S'$ has a compact support, we have
\begin{equation}\label{17}
\begin{aligned}
&{\partial b_{i,S}(u_i) \over \partial t}-
\mathop{\rm div}S'(u_i)a(x,u_i,Du_i))+S''(u_i)a(x,u_i,Du_i)Du_i\\
&- \mathop{\rm div}S'(u_i)\Phi_i(u_i))+S''(u_i)\Phi_i(u_i)Du_i +
f_i(x,u_1,u_2)S'(u_i)=0\quad \text{in } D'(Q),
\end{aligned}
\end{equation}
and
\begin{equation}\label{18}
b_{i,S}(u_i)(t=0)=b_{i,S}(u_{i,0})\quad \text{in } \Omega,
\end{equation}
where $b_{i,S}(r)=\int_0^r b_i'(s)S'(s)\,ds$.
\end{definition}
\begin{remark}\label{r2} \rm
Equation \eqref{17} is formally obtained through pointwise
multiplication of equation \eqref{1} by $S'(u_i)$. Note that in
Definition \ref{d1}, the gradient $Du_i$ is not defined even as a
distribution, but that due to \eqref{15} each term in \eqref{17} has
a meaning in $L^1(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$.
Indeed if $K>0$ is such that $\mathop{\rm supp} S' \subset [-K,K]$,
the following identifications are made in \eqref{17}:
\begin{itemize}
\item $ b_{i,S}(u_i)$ belong to $L^\infty(Q)\cap
L^p(0,T;W^{1,p}_0(\Omega))$. Indeed
$$
D b_{i,S}(u)=S'(u_i) b_i'(T_K(u_i))DT_K(u_i)\in (L^p(\Omega))^N
$$
and
$$ |b_{i,S}(u_i)| \leq \int_0^{|u_i|}
|S'(s)b_i'(s)|\,ds \leq K \max_{|r|\leq K} |S'(r)b_i'(r)|.
$$
\item $S'(u_i)a(x,u_i,Du_i)$ can be identified with
$S'(u_i)a(x,T_{K}(u_i),DT_{K}(u_i))$ a.e. in $Q$. Indeed, since
$|T_{K}(u_i)|\leq K$ a.e. in $Q$, assumptions \eqref{4} and
\eqref{6} imply that
$$
\big|a(x,T_{K}(u_i),DT_{K}(u_i))\big| \leq C_{K}(t,x) +
\beta_{K}|DT_{K}(u_i)|^{p-1}\quad \text{a.e. in } Q.
$$
As a consequence of \eqref{15} and of $S'(u_i) \in L^\infty(Q)$, it
follows that
$$
S'(u_i)a(x,T_{K}(u_i),DT_{K}(u_i)) \in (L^{p'}(Q))^N.
$$
\item $S''(u_i)a(x,u_i,Du_i)Du_i$ can be identified with
\[
S''(u_i)a(x,T_{K}(u_i),DT_{K}(u_i))DT_{K}(u_i)
\]
and in view of \eqref{4}, \eqref{6} and \eqref{15} one has
$$
S''(u_i)a(x,T_{K}(u_i),DT_{K}(u_i))DT_{K}(u_i) \in L^1(Q).
$$
\item $S'(u_i)\Phi_i(u_i)$ and $S''(u_i)\Phi_i(u_i)Du_i$
respectively identify with \\
$S'(u_i)\Phi_i(T_{K}(u_i))$ and
$S''(u_i)\Phi(T_{K}(u_i))DT_{K}(u_i)$. Due to the properties of $S$
and \eqref{8}, the functions $S', S''$ and $\Phi\circ T_{K}$ are
bounded on $\mathbb{R}$ so that \eqref{15} implies that
$S'(u_i)\Phi_i(T_{K}(u_i))\in (L^\infty(Q))^N$ and
$S''(u_i)\Phi_i(T_{K}(u_i))DT_{K}(u_i)$ belongs to $L^p(Q)$.
\item $S'(u_i)f_i(x,u_1,u_2)$ identifies with
$S'(u_i)f_1(x,T_K(u_1),u_2)$ a.e. in $Q$\\
(or $S'(u_i)f_2(x,u_1,T_K(u_2))$ a.e. in $Q$). Indeed, since
$|T_{K}(u_i)|\leq K$ a.e. in $Q$, assumptions \eqref{12} and
\eqref{13} imply that
$$
\big|f_1(x,T_K(u_1),u_2)\big| \leq F_K(x) + \sigma_K\
|b_2(u_2)|\quad \text{a.e. in } Q
$$
and
$$
\big|f_2(x,u_1,T_K(u_2))\big| \leq G_K(x) + \sigma_K\
|b_1(u_1)|\quad \text{a.e. in } Q.
$$
As a consequence of \eqref{15} and of $S'(u_i) \in
L^\infty(Q)$, it follows that
$$
S'(u_1)f_1(x,T_K(u_1),u_2) \in L^{1}(Q)\quad \text{and}\quad
S'(u_2)f_2(x,u_1,T_K(u_2)) \in L^{1}(Q).
$$
\end{itemize}
The above considerations show that \eqref{17} takes place in $D'(Q)$
and that ${\partial b_{i,S}(u_i) \over \partial t}$ belongs to
$L^{p'}(0,T;W^{-1,p'}(\Omega))+L^1(Q)$, which in turn implies that
${\partial b_{i,S}(u_i) \over \partial t}$ belongs to
$L^{1}(0,T;W^{-1,s}(\Omega))$ for all $s < inf(p',{N\over{N-1}})$.
It follows that $b_{i,S}(u_i)$ belongs to
$C^0([0,T];W^{-1,s}(\Omega))$ so that the initial condition
\eqref{18} makes sense.
\end{remark}
\section{Existence result}\label{s3}
This section is devoted to the proof of the following existence
theorem.
\begin{theorem}\label{t1}
Under assumptions \eqref{4}-\eqref{14}, there exists at least a
renormalized solution $(u_1, u_2)$ of Problem \eqref{1}-\eqref{3}.
\end{theorem}
\begin{proof}
The proof is divided into 9 steps. In step1, we introduce an
approximate problem and step 2 is devoted to establish a few {\it a
priori} estimates. In step 3, we prove some properties of the limit
$u_i$ of the approximate solutions $u^\varepsilon_i$. In step 4, we
define a time regularization of the field $T_K(u_i)$ and we
establish Lemma \ref{lem1} which allows
to control the parabolic contribution that arises in the
monotonicity method when passing to the limit. In step 5, we prove
an energy estimate (see Lemma \ref{lem2}) which is a key point for
the monotonicity arguments that are developed in Step 6 and Step 7.
In Step 8, we prove that $u_i$ satisfies \eqref{16} and finally, in
step 9, we prove that $u_i$ satisfies properties \eqref{17} and
\eqref{18} of Definition \ref{d1}.
\end{proof}
\noindent\textbf{Step 1.} Let us introduce the following
regularization of the data: for $\varepsilon>0$ and $i=1,2$
\begin{gather}\label{19}
b_{i,\varepsilon}(s)=b_i(T_{1\over\varepsilon}(s))+\varepsilon\
s\quad \forall s\in \mathbb{R},
\\ \label{20}
a_\varepsilon(x,s,\xi)=a(x,T_{1\over\varepsilon}(s),\xi)\ \text{a.e.
in}\ \Omega, \forall s\in \mathbb{R}, \forall \xi \in \mathbb{R}^N,
\\ \label{21}
\Phi_{i,\varepsilon} \text{ is a Lipschitz continuous bounded
function from $\mathbb{R}$ into $\mathbb{R}^N$}
\end{gather}
such that $\Phi_i^\varepsilon$ converges uniformly to $\Phi_i$ on
any compact subset of $\mathbb{R}$ as $\varepsilon$ tends to $0$.
\begin{gather}\label{22}
f_1^\varepsilon(x,s_1,s_2)
=f_1(x,T_{1\over\varepsilon}(s_1),T_{1\over\varepsilon}(s_2))\quad
\text{a.e. in } \Omega, \forall s_1, s_2\in \mathbb{R},
\\ \label{23}
f_2^\varepsilon(x,s_1,s_2)
=f_2(x,T_{1\over\varepsilon}(s_1),T_{1\over\varepsilon}(s_2))\quad
\text{a.e. in } \Omega, \forall s_1, s_2\in \mathbb{R},
\\ \label{24}
u_{i,0}^\varepsilon \in C^\infty_0(\Omega),
b_{i,\varepsilon}(u_{i,0}^\varepsilon)\to b_i(u_{i,0})\quad \text{in
} L^1(\Omega) \text{ as $\varepsilon$ tends to $0$}.
\end{gather}
Let us now consider the regularized problem
\begin{gather}\label{25}
{\partial b_{i,\varepsilon}(u^\varepsilon) \over \partial
t}- \mathop{\rm div}\big(a_\varepsilon(x,u^\varepsilon,D
u^\varepsilon)+\Phi_{i,\varepsilon}(u^\varepsilon)\big)+
f_i^\varepsilon(x,u_1^\varepsilon,u_2^\varepsilon)= 0 \quad \text{
in } Q,
\\ \label{26}
u_i^\varepsilon=0\quad \text{on } (0,T)\times{\partial\Omega},
\\ \label{27}
b_{i,\varepsilon}(u_i^\varepsilon)(t=0)
=b_{i,\varepsilon}(u_{i,0}^\varepsilon)\quad\text{in } \Omega.
\end{gather}
In view of \eqref{9} and \eqref{19}, for $i=1,2$, we have
$$
b'_{i,\varepsilon}(s)\geq \varepsilon, \quad
|b_{i,\varepsilon}(s)| \leq \max_{|s|\leq {1\over
\varepsilon}}|b_i(s)|+1 \quad \forall s\in \mathbb{R},
$$
In view of \eqref{6}, \eqref{12} and \eqref{13}, $a_\varepsilon,
f_1^\varepsilon$ and $f_2^\varepsilon$ satisfy: There exists
$C_\varepsilon \in L^{p'}(\Omega), F_\varepsilon\in L^1(\Omega),
G_\varepsilon\in L^1(\Omega)$ and $\beta_\varepsilon>0,
\sigma_\varepsilon>0, \lambda_\varepsilon>0$, such that
\begin{gather*}
|a_\varepsilon(x,s,\xi)| \leq C_\varepsilon(x)+\beta_\varepsilon
|\xi|^{p-1}\quad\text{a.e. in } x\in \Omega, \forall s\in
\mathbb{R}, \forall \xi \in \mathbb{R}^N.
\\
|f^\varepsilon_1(x,s_1,s_2)|\leq F_\varepsilon(x)+\sigma_\varepsilon\
\max_{|s|\leq {1\over \varepsilon}}|b_i(s)| \quad \text{a.e. in }
x\in \Omega, \forall s_1, s_2\in \mathbb{R},
\\
|f^\varepsilon_2(x,s_1,s_2)|\leq
G_\varepsilon(x)+\lambda_\varepsilon\ \max_{|s|\leq {1\over
\varepsilon}}|b_i(s)|\quad \text{a.e. in }x\in \Omega, \forall s_1,
s_2\in \mathbb{R}.
\end{gather*}
As a consequence, proving the existence of a weak solution
$u_i^\varepsilon \in L^p(0,T;W^{1,p}_0(\Omega))$ of
\eqref{25}-\eqref{27} is an easy task (see e.g. \cite{E1,H1,H2}).
\noindent\textbf{Step 2.} The estimates derived in this step rely
on usual techniques for problems of type \eqref{27}-\eqref{31} and
we just sketch the proof of them (the reader is referred to
\cite{B,BM,BR,BG,BMR,BMR1} or to \cite{BGDM,M,FM} for elliptic
versions of \eqref{27}-\eqref{31}).
Using $T_K(u_i^\varepsilon)$ as a test function in \eqref{25} leads
to
\begin{equation}\label{28}
\begin{aligned}
&\int_\Omega b_{i,\varepsilon}^K(u_i^\varepsilon)(t)\,dx +\int_0^t
\int_\Omega a_\varepsilon(x,u_i^\varepsilon,
Du_i^\varepsilon)DT_K(u_i^\varepsilon)\,dx\,ds \\
&+ \int_0^t \int_\Omega \Phi_{i,\varepsilon}(u_i^\varepsilon)
DT_K(u_i^\varepsilon)\,dx\,ds+\int_0^t \int_\Omega
f_i^\varepsilon(x, u_1^\varepsilon,
u_2^\varepsilon)T_K(u_i^\varepsilon)\,dx\,ds \\
&=\int_\Omega b_{i,\varepsilon}^K(u_{i,0}^\varepsilon)\,dx
\end{aligned}
\end{equation}
for $i=1,2$, for almost every $t$ in $(0,T)$, and where $
b_{i,\varepsilon}^K(r)= \int_0^r T_K(s) b_{i,\varepsilon}'(s)\,ds$.
The Lipschitz character of $\Phi_{i,\varepsilon}$, Stokes formula
together with the boundary condition \eqref{26} allow to obtain
obtain
\begin{equation}\label{29}
\int_0^t \int_\Omega
\Phi_{i,\varepsilon}(u_i^\varepsilon)DT_K(u_i^\varepsilon)\,dx\,ds
=0,
\end{equation}
for almost any $t \in (0,T)$. Now, as $ 0\leq
b_{i,\varepsilon}^K(u_{i,0}^\varepsilon)\leq
K|b_{i,\varepsilon}(u_{i,0}^\varepsilon)|
\quad\text{a.e. in}\ \Omega$, it follows that
$ 0\leq \int_\Omega
b_{i,\varepsilon}^K(u_{i,0}^\varepsilon)\,dx\leq K\int_\Omega
|b_{i,\varepsilon}(u_{i,0}^\varepsilon)|\,dx$. Since $a_\varepsilon$
satisfies \eqref{20}, $f_i^\varepsilon$ satisfies \eqref{22},
\eqref{23}, we deduce from \eqref{32} ( taking into account the
properties of $ b_{i,\varepsilon}^K$ and $u_{i,0}^\varepsilon$ )
that
\begin{equation}\label{30}
T_K(u_i^\varepsilon)\text{ is bounded in }
L^p(0,T;W^{1,p}_0(\Omega))
\end{equation}
independently of $\varepsilon$ for any $K\geq 0$.
Proceeding as in \cite{BM,BR,BMR}, we prove
that for any $S \in W^{2,\infty}(\mathbb{R})$ such that $S'$
is compact ($\mathop{\rm supp} S'\subset [-K,K]$)
\begin{equation}\label{31}
S(b_{i,\varepsilon}(u_i^\varepsilon)) \text{ is bounded in }
L^p(0,T;W^{1,p}_0(\Omega)),
\end{equation}
and
\begin{equation}\label{32}
{\partial S(b_{i,\varepsilon}(u_i^\varepsilon)) \over \partial t}
\text{ is bounded in } L^1(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega)),
\end{equation}
independently of $\varepsilon$, as soon as $\varepsilon0:a_\varepsilon(x,T_{K}(u_i^\varepsilon),
DT_{K}(u_i^\varepsilon))=
a(x,T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon))$ a.e. in $Q$ as
long as $\varepsilon0$ and $C_K \in L^{p'}(Q)$. In view of \eqref{30}, we
deduce that
\begin{equation}\label{35}
a\big(x,T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon)\big)
\text{ is bounded in } (L^{p'}(Q))^N.
\end{equation}
independently of $\varepsilon$ for $\varepsilon0$ and any $n\geq 1$. Here,
for any $K>0$ and for $i=1,2, X_{i,K}$ belongs to $(L^{p'}(Q))^N$.
We now establish that $b_i(u_i)$ belongs to
$L^\infty(0,T;L^1(\Omega))$. Indeed using
${1\over\sigma}T_{\sigma}(u_i^\varepsilon)$ as a test function in
\eqref{25} leads to
\begin{equation} \label{43}
\begin{aligned}
&{1\over \sigma}\int_\Omega
b_{i,\varepsilon}^\sigma(u_i^\varepsilon)(t)\,dx+{1\over
\sigma}\int_0^t \int_\Omega
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon) D
T_{\sigma}(u_i^\varepsilon)\,dx\,ds
\\
&+ {1\over \sigma} \int_0^t \int_\Omega
\Phi_{i,\varepsilon}(u_i^\varepsilon)D
T_{\sigma}(u_i^\varepsilon)\,dx\,ds+{1\over \sigma}\int_0^t
\int_\Omega f_i^\varepsilon(x,u_1^\varepsilon,u_2^\varepsilon)
T_{\sigma}(u_i^\varepsilon)\,dx\,ds \\
&= {1\over \sigma}\int_\Omega
b_{i,\varepsilon}^\sigma(u_{i,0}^\varepsilon)\,dx,
\end{aligned}
\end{equation}
for almost any $t$ in $(0,T)$. Where,
$b_{i,\varepsilon}^n(r)=\int_0^r b'_{i,\varepsilon}(s)
T_{\sigma}(s)\,ds$.
The Lipschitz character of $\Phi_\varepsilon$, Stokes formula
together with the boundary condition \eqref{26} allow to obtain
\begin{equation}\label{44}
{1\over \sigma}\int_0^t \int_\Omega
\Phi_{i,\varepsilon}(u_i^\varepsilon)D
T_{\sigma}(u_i^\varepsilon)\,dx\,ds= 0.
\end{equation}
Since $a_\varepsilon$ satisfies \eqref{5} and $f_i^\varepsilon$
satisfies \eqref{11}, letting $\sigma$ go to zero, it follows that
\begin{equation}\label{45}
\int_\Omega |b_{i,\varepsilon}(u_i^\varepsilon)(t)|\,dx\leq
\|b_{i,\varepsilon}(u_{i,0}^\varepsilon)\|_{L^1(\Omega)}
\end{equation}
for almost $t \in (0,T)$. Recalling \eqref{24}, \eqref{39} and
\eqref{45} makes it possible to pass to the limit-inf and we show
that $b_i(u_i)$ belongs to $L^{\infty}(0,T;L^1(\Omega))$.
We are now in a position to exploit \eqref{38}. The pointwise
convergence of $u^\varepsilon$ to $u$
and $b_{i,\varepsilon}(u^\varepsilon_0)$ to $b_i(u_0)$ imply that
\begin{equation}\label{46}
{\limsup_{\varepsilon \to 0}}\int_0^t \int_\Omega
a(x,u_i^\varepsilon,Du_i^\varepsilon)D\theta_n(u_i^\varepsilon)\,dx\,ds\leq
\int_\Omega b_i^n(u_{i,0})\,dx,
\end{equation}
Since $\theta_n$ converge to zero everywhere as $n$ goes to zero,
the Lebesgue's convergence theorem permits to conclude that
\begin{equation}\label{47}
\lim_{n\to +\infty} {\limsup_{\varepsilon \to 0}}
\int_{\{n \leq |u_i^\varepsilon|\leq n+1\}}
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon\,dx\,dt
=0.
\end{equation}
\noindent\textbf{Step 4.} This step is devoted to introduce for
$K\geq 0$ fixed, a time regularization of the function $T_K(u_i)$ in
order to perform the monotonicity method which will be developed in
Step 5 and Step 6. This kind of regularization has been first
introduced by Landes (see Lemma 6 and Proposition 3, p. 230 and
Proposition 4, p. 231 in \cite{L}). More recently, it has been
exploited in \cite{BDGO} and
\cite{DO} to solve a few nonlinear evolution problems with $L^1$
or measure data.
This specific time regularization of $T_K(u_i)$ (for fixed $K\geq
0$) is defined as follows. let us consider the unique solution
$T_K(u_i)_\mu \in L^\infty(Q)\cap L^p(0,T;W^{1,p}_0(\Omega))$ of the
monotone problem:
\begin{gather}\label{48}
{\partial T_K(u_i)_\mu \over \partial t} + \mu \Big(T_K(u_i)_\mu -
T_K(u_i)\Big)=0 \quad \text{in } D'(Q).
\\ \label{49}
T_K(u_i)_\mu(t=0)= 0 \ \text{in } \Omega.
\end{gather}
We remark that for $\mu>0$ and $K\geq 0$,
\begin{equation}\label{50}
{\partial T_K(u_i)_\mu \over \partial t} \in
L^p(0,T;W^{1,p}_0(\Omega)).
\end{equation}
The behavior of $T_K(u_i)_\mu$ as $\mu \to +\infty$ is investigated
in \cite{L} (see also \cite{DO} and \cite{G}) and we just recall
here that \eqref{48}-\eqref{49} imply that
\begin{equation}\label{51}
T_K(u_i)_\mu\to T_K(u_i) \quad \text{a.e. in } Q\,,
\end{equation}
and in $L^\infty(Q)$ weak $\star$ and strongly in
$L^p(0,T;W^{1,p}_0(\Omega))$ as $\mu\to +\infty$.
\begin{equation}\label{52}
\|T_K(u_i)_\mu\|_{L^\infty(Q)} \leq K
\end{equation}
for any $\mu$ and any $K\geq 0$.
Let $v_{i,j}\in C^\infty_0(\Omega)$, such that $v_{i,j}$ converges
almost everywhere to $u_{i,0}$ in $\Omega$. And let us consider
$$
{T_K(u_i)}_{\mu,j}={T_K(u_i)}_\mu + e^{-\mu t}T_K(v_{i,j})
$$
is a smooth approximation of $T_K(u_i)$. We remark that for $\mu>0,
j>0$ and $K\geq 0$, we have $|T_K(u_i)_{\mu,j}|\leq K$ and
\begin{gather}\label{53}
{\partial T_K(u_i)_{\mu,j} \over \partial t} =
\mu\Big(T_K(u_i)-T_K(u_i)_{\mu,j}\Big),
\\ \label{54}
T_K(u_i)_{\mu,j}(0)=T_K(v_{i,j}),
\\ \label{55}
T_K(u_i)_{\mu,j}\to T_K(u_i)\quad \mbox{strongly in}\
L^p(0,T;W^{1,p}_0(\Omega)),
\end{gather}
as $\mu$ tends to infinity.
We denote by $w(\varepsilon,\mu,j)$ the quantities such that
$$
\lim_{j\to+\infty} \lim_{\mu\to+\infty} \lim_{\varepsilon\to 0}
w(\varepsilon,\mu,j)=0.
$$
The main estimate is as follows.
\begin{lemma}\label{lem1}
Let $K\geq 0$ be fixed. Let $S$ be an increasing
$C^\infty(\mathbb{R})$-function such that $S(r)=r$ for $|r|\leq K$
and $\mathop{\rm supp}(S')$ is compact. Then
$$
{ {\liminf_{\mu\to +\infty}}}\lim_{\varepsilon\to 0 }\int_0^T
\int_0^s {\big\langle{\partial b_{i,S}(u_i^\varepsilon) \over
\partial t}, \big(T_K(u_i^\varepsilon) - (T_K(u_i))_\mu\big)\big\rangle}\,dt\,ds
\geq 0
$$
where $\langle, \rangle$ denotes the duality pairing between
$L^1(\Omega)+W^{-1,p'}(\Omega)$ and $L^\infty(\Omega)\cap
W^{1,p}_0(\Omega)$. and where $ b_{i,S}(r)=\int_0^r
b_i'(s)S(s)\,ds$.
\end{lemma}
The proof of the above Lemma can be found in \cite{R1}.
\noindent\textbf{Step 5.}
In this step we prove
the following Lemma which is the key point in the monotonocity
arguments that will be developed in Step 6.
\begin{lemma}\label{lem2}
The subsequence of $u^\varepsilon$ defined is Step 3 satisfies: For
any $K\geq 0$,
\begin{equation}\label{56}
\begin{aligned}
&{{\limsup_{\varepsilon\to 0}}} \int_0^T \int_0^t \int_\Omega
a(u_i^\varepsilon,DT_{K}(u_i^\varepsilon))
DT_K(u_i^\varepsilon)\,dx\,ds\,dt\\
&\leq \int_0^T \int_0^t \int_\Omega X_{i,K}DT_K(u_i)\,dx\,ds\,dt
\end{aligned} \end{equation}
\end{lemma}
\begin{proof}
We first introduce a
sequence of increasing $C^\infty(\mathbb{R})$-functions $S_n$ such
that, for any $n\geq 1$
\begin{equation}\label{57}
S_n(r)=r \text{ for } |r|\leq n, \quad \mathop{\rm supp}( S'_n)
\subset [-(n+1),(n+1)], \quad \|S_n''\|_{L^\infty(\mathbb{R})} \leq
1.
\end{equation}
Pointwise multiplication of \eqref{25} by $S'_n(u_i^\varepsilon)$
(which is licit) leads to
\begin{equation} \label{60}
\begin{aligned}
&{\partial b_{i,{S_n}}(u_i^\varepsilon) \over \partial t}-
\mathop{\rm div}\Big(S_n(u_i^\varepsilon)
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)\Big) +
S''_n(u_i^\varepsilon)a_\varepsilon(x,u_i^\varepsilon,
Du_i^\varepsilon)Du_i^\varepsilon \\
&- \mathop{\rm div}\Big(\Phi_{i,\varepsilon}(u_i^\varepsilon)
S'_n(u_i^\varepsilon)\Big)
+S''_n(u_i^\varepsilon)\Phi_{i,\varepsilon}(u_i^\varepsilon)
Du_i^\varepsilon+ f^\varepsilon_i(x,u_1^\varepsilon,u_2^\varepsilon)
S_n'(u_i^\varepsilon)=0
\end{aligned}
\end{equation}
in $D'(Q)$. We use the sequence $T_K(u)_\mu$ of approximations of
$T_K(u)$ defined by \eqref{48}, \eqref{49} of Step 4 and plug the
test function $T_K(u^\varepsilon)-T_K(u)_\mu$ (for $\varepsilon>0$
and $\mu>0$) in \eqref{60}. Through setting, for fixed $K\geq 0$,
\begin{equation}\label{61}
W_{i,\mu}^\varepsilon = T_K(u_i^\varepsilon)-T_K(u_i)_\mu
\end{equation}
we obtain upon integration over $(0,t)$ and then over $(0,T)$,
\begin{equation} \label{62}
\begin{aligned}
&\int_0^T \int_0^t \big\langle {\partial
b_{i,{S_n}}(u_i^\varepsilon) \over
\partial t}, W_{i,\mu}^\varepsilon\big\rangle\,ds \,dt\\
&+\int_0^T\int_0^t \int_\Omega S'_n(u_i^\varepsilon)
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)
DW_{i,\mu}^\varepsilon\,dx\,ds\,dt
\\
&+\int_0^T\int_0^t \int_\Omega S''_n(u_i^\varepsilon)
W_{i,\mu}^\varepsilon
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)
Du_i^\varepsilon\,dx\,ds \,dt
\\
&+\int_0^T \int_0^t \int_\Omega
\Phi_{i,\varepsilon}(u_i^\varepsilon)S'_n(u_i^\varepsilon)
DW_{i,\mu}^\varepsilon\,dx\,ds\,dt
\\
&+\int_0^T \int_0^t \int_\Omega
S''_n(u_i^\varepsilon)W_{i,\mu}^\varepsilon
\Phi_{i,\varepsilon}(u_i^\varepsilon)Du_i^\varepsilon\,dx\,ds\,dt
\\
&+ \int_0^T \int_0^t \int_\Omega
f^\varepsilon_i(x,u_1^\varepsilon,u_2^\varepsilon)
S'_n(u_i^\varepsilon) W_{i,\mu}^\varepsilon\,dx\,ds\,dt=0
\end{aligned}
\end{equation}
Next we pass to the limit as $\varepsilon$ tends to $0$, then $\mu$
tends to $+\infty$ and then $n$ tends to $+\infty$, the real number
$K\geq 0$ being kept fixed. In order to perform this task we prove
below the following results for fixed $K\geq 0$:
\begin{gather}\label{63}
{{{\liminf_{\mu \to +\infty}}}} \lim_{\varepsilon \to 0} \int_0^T
\int_0^t \big\langle {\partial b_{i,{S_n}}(u_i^\varepsilon) \over
\partial t}\ , W_{i,\mu}^\varepsilon\big\rangle\,ds \,dt \geq 0\quad
\text{for any } n\geq K,
\\ \label{64}
\lim_{\mu \to +\infty}\lim_{\varepsilon \to 0}\int_0^T \int_0^t
\int_\Omega
S'_n(u_i^\varepsilon) \Phi_{i,\varepsilon}(u_i^\varepsilon)
DW_{i,\mu}^\varepsilon\,dx\,ds\,dt=0 \quad\text{for any } n\geq 1,
\\ \label{65}
\lim_{\mu \to +\infty}\lim_{\varepsilon \to 0} \int_0^T \int_0^t
\int_\Omega S''_n(u_i^\varepsilon) W_{i,\mu}^\varepsilon
\Phi_{i,\varepsilon}(u_i^\varepsilon)Du_i^\varepsilon\,dx\,ds\,dt = 0
\quad \text{for any } n,
\\ \label{66}
\lim_{n \to +\infty}{\overline{\lim_{\mu \to +\infty}}}\
{\overline{\lim_{\varepsilon \to 0}}}\big|\int_0^T\int_0^t
\int_\Omega S''_n(u_i^\varepsilon) W_{i,\mu}^\varepsilon
a_\varepsilon(u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon
\,dx\,ds\,dt\big|=0,
\\
\label{67} \lim_{\mu \to +\infty}\lim_{\varepsilon \to 0}\int_0^T
\int_0^t \int_\Omega
f^\varepsilon_i(x,u_1^\varepsilon,u_2^\varepsilon)
S'_n(u_i^\varepsilon) W_{i,\mu}^\varepsilon\,dx\,ds\,dt=0 \quad
\text{for any } n\geq 1.
\end{gather}
\end{proof}
\textbf{Proof of \eqref{63}.} In view of
\eqref{61} of $W_{i,\mu}^\varepsilon$, Lemma
\ref{lem1} applies with $S=S_n$ for fixed $n\geq K$. As a
consequence \eqref{63} holds.
\textbf{Proof of \eqref{64}.} For fixed $n\geq 1$, we have
\begin{equation}\label{68}
S_n'(u_i^\varepsilon)\Phi_{i,\varepsilon}(u_i^\varepsilon)
DW^\varepsilon_{i,\mu}
=S_n'(u_i^\varepsilon)\Phi_{i,\varepsilon}(T_{n+1}(u_i^\varepsilon))
DW^\varepsilon_{i,\mu}
\end{equation}
a.e. in $Q$, and for all $\varepsilon\leq {1\over{n+1}}$, and where
$\mathop{\rm supp} S_n'\subset[-(n+1),n+1]$.
Since $S'_n$ is smooth and bounded, \eqref{8}, \eqref{23} and
\eqref{40} lead to
\begin{equation}\label{69}
S_n'(u_i^\varepsilon)\Phi_{i,\varepsilon}(T_{n+1}(u_i^\varepsilon))\to
S_n'(u_i)\Phi_i(T_{n+1}(u_i))
\end{equation}
a.e. in $Q$ and in $L^\infty(Q)$ weak $\star$, as $\varepsilon$
tends to $0$. For fixed $\mu>0$, we have
\begin{equation}\label{70}
W_{i,\mu}^\varepsilon \rightharpoonup T_K(u_i)-T_K(u_i)_\mu\quad
\text{weakly in }L^p(0,T;W^{1,p}_0(\Omega))
\end{equation}
and a.e. in $Q$ and in $L^\infty(Q)$ weak $\star$, as $\varepsilon$
tends to $0$. As a consequence of \eqref{68}, \eqref{69} and
\eqref{70} we deduce that
\begin{equation} \label{71}
\begin{aligned}
&\lim_{\varepsilon\to 0}\int_0^T \int_0^t \int_\Omega
S_n'(u_i^\varepsilon)
\Phi_{i,\varepsilon}(u_i^\varepsilon)DW^\varepsilon_{i,\mu}\,dx\,ds\,dt
\\
&=\int_0^T \int_0^t \int_\Omega S_n'(u_i)\Phi_i(u_i)\big[DT_K(u_i)-
DT_K(u_i)_\mu\big]\,dx\,ds\,dt
\end{aligned}
\end{equation}
for any $\mu>0$. Appealing now to \eqref{51} and passing to the
limit as $\mu \to +\infty$ in \eqref{71} allows to conclude that
\eqref{64} holds.
\textbf{Proof of \eqref{65}.} For fixed $n\geq 1$, and by the same
arguments as those which lead to \eqref{68}, we have
\begin{equation*}
S_n''(u_i^\varepsilon)\Phi_{i,\varepsilon}(u_i^\varepsilon)
Du_i^\varepsilon W^\varepsilon_{i,\mu} =
S_n''(u_i^\varepsilon)\Phi_{i,\varepsilon}(T_{n+1}(u_i^\varepsilon))
DT_{n+1}(u_i^\varepsilon) W^\varepsilon_{i,\mu} \quad \text{a.e. in
} Q.
\end{equation*}
From \eqref{8}, \eqref{21} and \eqref{40}, it follows that for any
$\mu>0$,
\begin{align*}
&\lim_{\varepsilon\to 0}\int_0^T \int_0^t \int_\Omega
S_n''(u_i^\varepsilon)
\Phi_{i,\varepsilon}(u_i^\varepsilon)Du_i^\varepsilon
W^\varepsilon_{i,\mu}\,dx\,ds\,dt
\\
&=\int_0^T \int_0^t \int_\Omega S_n''(u_i)\Phi_{i}(T_{n+1}(u_i))
DT_{n+1}(u_i) W_{i,\mu}\big[DT_K(u_i)-
DT_K(u_i)_\mu\big]\,dx\,ds\,dt
\end{align*}
with the help of \eqref{55} passing to the limit, as $\mu$ tends to
$+\infty$, in the above equality, we find \eqref{65}.
\textbf{Proof of \eqref{66}.} For any $n\geq 1$ fixed, we have
$\mathop{\rm supp} (S''_n)\subset [-(n+1),-n]\cup [n,n+1]$. As a
consequence
\begin{align*}
&\big|\int_0^T \int_0^t \int_\Omega S_n''(u_i^\varepsilon)
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon
W^\varepsilon_{i,\mu}\,dx\,ds\,dt \big|
\\
&\leq T \|S''_n\|_{L^\infty(\mathbb{R})}
\|W^\varepsilon_{i,\mu}\|_{L^\infty(Q)} \int_{\{ n \leq
|u_i^\varepsilon|\leq {n+1}\}}
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)
Du_i^\varepsilon\,dx\,dt,
\end{align*}
for any $n\geq 1$, and any $\mu>0$. The above inequality together
with \eqref{52} and \eqref{57} make it possible to obtain
\begin{equation} \label{72}
\begin{aligned}
&{\limsup_{\mu \to +\infty}} {\limsup_{\varepsilon \to 0}}
\big|\int_0^T \int_0^t \int_\Omega S_n''(u_i^\varepsilon)
a_\varepsilon(u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon
W^\varepsilon_{i,\mu}\,dx\,ds\,dt \big|
\\
&\leq C {\limsup_{\varepsilon \to 0}} \int_{\{ n \leq
|u_i^\varepsilon|\leq {n+1}\}}
a_\varepsilon(u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon\,dx\,dt,
\end{aligned}
\end{equation}
for any $n\geq 1$, where $C$ is a constant independent of $n$. Using
\eqref{47} we pass to the limit as $n$ tends to $+\infty$ in
\eqref{72} and establish \eqref{66}.
\textbf{Proof of \eqref{67}.} For fixed $n\geq 1$, we have,
\begin{gather*}
f^\varepsilon_1(x,u_1^\varepsilon,u_2^\varepsilon)
S_n'(u_1^\varepsilon)=f_1(x,T_{n+1}(u_1^\varepsilon),T_{1\over\varepsilon
}(u_2^\varepsilon))S_n'(u_1^\varepsilon),
\\
f^\varepsilon_2(x,u_1^\varepsilon,u_2^\varepsilon)
S_n'(u_2^\varepsilon)=f_2(x,T_{1\over\varepsilon}(u_1^\varepsilon),
T_{n+1}(u_2^\varepsilon))S_n'(u_2^\varepsilon)
\end{gather*}
a.e. in $Q$, and for all $\varepsilon\leq {1\over{n+1}}$. In view
of \eqref{10}, \eqref{39} and \eqref{40}, Lebesgue's convergence
theorem implies that for any $\mu>0$ and any $n\geq 1$
\begin{align*}
&\lim_{\varepsilon \to 0}\int_0^T \int_0^t \int_\Omega
f^\varepsilon_1(x,u_1^\varepsilon,u_2^\varepsilon)
S_n'(u_i^\varepsilon)W^\varepsilon_\mu\,dx\,ds\,dt
\\
&= \int_0^T \int_0^t \int_\Omega f_1(x,u_1,u_2)
S'_n(u_i)\Big(T_K(u_i)- T_K(u_i)_\mu\Big)\,dx\,ds\,dt.
\end{align*}
Now for fixed $n\geq 1$, using \eqref{51} permits to pass to the
limit as $\mu$ tends to $+\infty$ in the above equality to obtain
\eqref{67}.
We now turn back to the proof of Lemma \ref{lem2}, due to
\eqref{63}, \eqref{64}, \eqref{65}, \eqref{66} and \eqref{67}, we
are in a position to pass to the lim-sup when $\varepsilon$ tends to
zero, then to the limit-sup when $\mu$ tends to $+\infty$ and then
to the limit as $n$ tends to $+\infty$ in \eqref{62}. We obtain
using the definition of $W^\varepsilon_\mu$ that for any $K\geq 0$,
\begin{align*}
&\lim_{n \to +\infty}{\limsup_{\mu \to +\infty}}
{\limsup_{\varepsilon \to 0}}
\int_0^T \int_0^t \int_\Omega S_n'(u_i^\varepsilon)
a_\varepsilon(u_i^\varepsilon,Du_i^\varepsilon)
\big(DT_K(u_i^\varepsilon)\\
&-DT_K(u_i)_\mu\big)\,dx\,ds\,dt \leq 0.
\end{align*}
Since $S_n'(u_i^\varepsilon)a_\varepsilon(u_i^\varepsilon,
Du_i^\varepsilon)DT_K(u_i^\varepsilon)
=a(u_i^\varepsilon,Du_i^\varepsilon)DT_K(u_i^\varepsilon)$ for
$\varepsilon\leq {1\over {K}}$ and $K\leq n$.
The above inequality implies that for $K\leq n$,
\begin{equation} \label{76}
\begin{aligned}
&{\limsup_{\varepsilon \to 0}}\int_0^T \int_0^t \int_\Omega
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)
DT_K(u_i^\varepsilon)\,dx\,ds\,dt
\\
&\leq \lim_{n \to +\infty}{\limsup_{\mu \to +\infty}}
{\limsup_{\varepsilon \to 0}} \int_0^T \int_0^t \int_\Omega
S'_n(u_i^\varepsilon)a_\varepsilon(x,u_i^\varepsilon,
Du_i^\varepsilon)DT_K(u_i)_\mu\,dx\,ds\,dt
\end{aligned}
\end{equation}
The right hand side of \eqref{76} is computed as follows: In view of
\eqref{20} and \eqref{60}, we have for $\varepsilon\leq {1\over
{n+1}}$,
\begin{equation*}
S'_n(u_i^\varepsilon)a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)=
S'_n(u_i^\varepsilon)a\Big(x,T_{n+1}(u_i^\varepsilon),DT_{n+1}(u_i^\varepsilon)\Big)\
\text{a.e. in}\ Q.
\end{equation*}
Due to \eqref{42}, it follows that for fixed $n\geq 1$,
\begin{equation*}
S'_n(u_i^\varepsilon)a_\varepsilon(u_i^\varepsilon,
Du_i^\varepsilon)\rightharpoonup S'_n(u_i)X_{i,{n+1}}\ \text{weakly
in}\ (L^{p'}(Q))^N,
\end{equation*}
when $\varepsilon$ tends to $0$.
The strong convergence of $T_K(u_i)_\mu$ to $T_K(u_i)$ in
$L^p(0,T;W^{1,p}_0(\Omega))$ as $\mu$ tends to $+\infty$, allows
then to conclude that
\begin{equation} \label{77}
\begin{aligned}
&\lim_{\mu \to +\infty} \lim_{\varepsilon \to 0} \int_0^T \int_0^t
\int_\Omega S'_n(u_i^\varepsilon)a_\varepsilon(x,u_i^\varepsilon,
Du_i^\varepsilon)DT_K(u_i)_\mu\,dx\,ds\,dt
\\
&= \int_0^T \int_0^t \int_\Omega
S'_n(u_i)X_{i,n+1}DT_K(u_i)\,dx\,ds\,dt\\
&= \int_0^T \int_0^t \int_\Omega X_{i,n+1}DT_K(u_i)\,dx\,ds\,dt
\end{aligned}
\end{equation}
as long as $K\leq n$, since $S'_n(r)=1$ for $|r|\leq n$. Now for
$K\leq n$, we have
$$
a\big(x,T_{n+1}(u_i^\varepsilon),DT_{n+1}(u_i^\varepsilon)\big)\chi_{\{|u_i^\varepsilon|<
K\}} =a\big(x,T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon)\big)
\chi_{\{|u_i^\varepsilon|< K\}},
$$
a.e. in $Q$. Passing to the limit as $\varepsilon$ tends to $0$, we
obtain
\begin{equation}\label{78}
X_{i,n+1} \chi_{\{|u_i|< K\}}= X_{i,K} \chi_{\{|u_i|<
K\}}\quad \text{a.e. in } Q-\{|u_i|=K \} \text{ for } K\leq n.
\end{equation}
As a consequence of \eqref{78}, for $K\leq n$, we have
\begin{equation}\label{79}
X_{n+1}DT_K(u_i)= X_{K}DT_K(u_i)\quad \text{a.e. in } Q.
\end{equation}
Taking into account \eqref{76}, \eqref{77} and \eqref{79}, we
conclude that \eqref{56} holds true and the proof of Lemma
\ref{lem2} is complete.
\textbf{Step 6.} In this step, we prove the following monotonicity
estimate.
\begin{lemma}\label{lem3}
The subsequence of $u_i^\varepsilon$ defined in step 3 satisfies:
For any $K\geq 0$,
\begin{equation} \label{80}
\begin{aligned}
&\lim_{\varepsilon \to 0} \int_0^T \int_0^t \int_\Omega
\big[a(T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon))
-a(T_{K}(u_i^\varepsilon),DT_{K}(u_i))\big]
\\
&\times
\big[DT_{K}(u_i^\varepsilon)-DT_{K}(u_i)\big]\,dx\,ds\,dt=0\,.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Let $K\geq 0$ be fixed. The monotone character \eqref{7} of
$a(s,\xi)$ with respect to $\xi$ implies that
\begin{equation} \label{81}
\begin{aligned}
&\int_0^T \int_0^t \int_\Omega
\big[a(T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon))
-a(T_{K}(u_i^\varepsilon),DT_{K}(u_i))\big]
\\
&\times \big[DT_{K}(u_i^\varepsilon)-DT_{K}(u_i)\big] \,dx\,ds\,dt
\geq 0,
\end{aligned}
\end{equation}
In order to pass to the limit-sup as $\varepsilon$ tends to $0$ in
\eqref{81}, let us recall first that \eqref{4}, \eqref{6} and
\eqref{39} imply
\begin{equation*}
a(T_{K}(u_i^\varepsilon),DT_{K}(u_i))\to
a(T_{K}(u_i),DT_{K}(u_i))\quad \text{a.e. in } Q,
\end{equation*}
as $\varepsilon$ tends to $0$, and that
\begin{equation*}
\big|a(T_{K}(u_i^\varepsilon),DT_{K}(u_i))\big| \leq
C_{K}(t,x) + \beta_K|DT_{K}(u_i)|^{p-1}
\end{equation*}
a.e. in $Q$, uniformly with respect to $\varepsilon$. It follows
that when $\varepsilon$ tends to $0$,
\begin{equation}\label{82}
a\big(T_{K}(u_i^\varepsilon),DT_{K}(u_i)\big)\to
a\big(T_{K}(u_i),DT_{K}(u_i)\big)\quad \text{strongly in }
(L^{p'}(Q))^N.
\end{equation}
Using \eqref{56} of Lemma \ref{lem2}, \eqref{40}, \eqref{42} and
\eqref{82}, we can pass to the lim-sup as $\varepsilon$ tends to
zero in \eqref{81} to obtain \eqref{80} of Lemma \ref{lem3}.
\end{proof}
\textbf{Step 7.} In this step we identify the weak limit $X_{i,K}$
and we prove the weak $L^1$ convergence of the ``truncated'' energy
$a\big(T_{K}(x,u_i^\varepsilon),DT_{K}(u_i^\varepsilon)\big)
DT_{K}(u_i^\varepsilon)$ as $\varepsilon$ tends to $0$.
\begin{lemma}\label{lem4}
For fixed $K\geq 0$, as $\varepsilon$ tends to $0$, we have
\begin{equation}\label{83}
X_{i,K}=
a\big(x,T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon)\big)\quad
\text{a.e. in } Q.
\end{equation}
Also, as $\varepsilon$ tends to $0$,
\begin{equation}\label{84}
a\big(T_{K}(u_i^\varepsilon),DT_{K}(u_i^\varepsilon)\big)
DT_{K}(u_i^\varepsilon)\rightharpoonup
a\big(T_{K}(u_i),DT_{K}(u_i)\big)DT_{K}(u_i),
\end{equation}
weakly in $L^1(Q)$.
\end{lemma}
\begin{proof}
The proof is standard once we remark that for any $K\geq 0$, any
$00$, the function $a_{1\over {K}}(x,s,\xi)$ is
continuous and bounded with respect to $s$, the usual Minty's
argument applies in view of \eqref{40}, \eqref{42}, and \eqref{85}.
It follows that \eqref{83} holds true (the case $K=0$ being
trivial). In order to prove \eqref{84}, we observe that thanks to
the monotone character of $a$ (with respect to $\xi$) and
\eqref{80}, for any $K\geq 0$ and any $T'T$, \eqref{6}-\eqref{14} are
satisfied with $\overline{T}$ in place of $T$ and that the
convergence result \eqref{87} is still true in $L^1(Q)$-weak which
means that \eqref{84} holds.
\textbf{Step 8.} In this step we prove that $u$ satisfies
\eqref{16}. To this end, we remark that for any fixed $n\geq 0$,
\begin{align*}
&\int_{\{(t,x)/\ n\leq |u_i^\varepsilon|\leq {n+1}\}}
a(x,u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon\,dx\,dt
\\
&=\int_Q
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)\Big[DT_{n+1}(u_i^\varepsilon)-
DT_{n}(u_i^\varepsilon)\Big]\,dx\,dt
\\
&=\int_Q
a_\varepsilon\Big(x,T_{n+1}(u_i^\varepsilon),DT_{n+1}(u_i^\varepsilon)\Big)D
T_{n+1}(u_i^\varepsilon)\,dx\,dt
\\
&\quad -\int_Q
a_\varepsilon\Big(x,T_{n}(u_i^\varepsilon),DT_{n}(u_i^\varepsilon)\Big)DT_{n}(u^\varepsilon)\,dx\,dt
\end{align*}
for $\varepsilon< {1\over {n+1}}$.
According to \eqref{84}, one can pass to the limit as
$\varepsilon$ tends to $0$; for fixed $n\geq 0$ to obtain
\begin{equation}\label{88}
\begin{aligned}
&\lim_{\varepsilon\to 0}\int_{\{(t,x)/\ n\leq |u_i^\varepsilon|\leq
{n+1}\}}
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)Du_i^\varepsilon\,dx\,dt
\\
&=\int_Q
a\Big(x,T_{n+1}(u_i),DT_{n+1}(u_i)\Big)DT_{n+1}(u_i)\,dx\,dt
\\
&\quad -\int_Q
a\Big(x,T_{n}(u_i),DT_{n}(u_i)\Big)DT_{n}(u_i)\,dx\,dt
\\
&=\int_{\{(t,x)/\ n\leq |u_i|\leq {n+1}\}}a(x,u_i,Du_i)Du_i\,dx\,dt
\end{aligned}
\end{equation}
Taking the limit as $n$ tends to $+\infty$ in \eqref{88} and using
the estimate \eqref{47} show that $u_i$ satisfies \eqref{16}.
\textbf{Step 9.} In this step, $u_i$ is shown to satisfy \eqref{17}
and \eqref{18}. Let $S$ be a function in $W^{2,\infty}(\mathbb{R})$
such that $S'$ has a compact support. Let $K$ be a positive real
number such that $\mathop{\rm supp} S'\subset [-K,K]$. Pointwise
multiplication of the approximate equation \eqref{25} by
$S'(u_i^\varepsilon)$ leads to
\begin{equation} \label{89}
\begin{aligned}
&{\partial b_{i,S}^\varepsilon(u_i^\varepsilon) \over \partial t} -
\mathop{\rm div}\Big(S'(u_i^\varepsilon)
a_\varepsilon(x,u_i^\varepsilon,Du_i^\varepsilon)\Big)
+S''(u_i^\varepsilon)a_\varepsilon(x,u_i^\varepsilon,
Du_i^\varepsilon)Du_i^\varepsilon
\\
&-\mathop{\rm div}\Big(S'(u_i^\varepsilon)\Phi_{i,\varepsilon}
(u_i^\varepsilon)\Big) +S''(u_i^\varepsilon)
\Phi_\varepsilon(u_i^\varepsilon)Du_i^\varepsilon +
f^\varepsilon_i(x,u_1^\varepsilon,u_2^\varepsilon)
S'(u_i^\varepsilon)=0
\end{aligned}
\end{equation}
in $D'(Q)$, for $i=1,2$. In what follows we pass to the limit as
$\varepsilon$ tends to $0$ in each term of \eqref{89}.
\textbf{Limit of ${\partial b_{i,S}^\varepsilon(u_i^\varepsilon)
\over \partial t}$.} Since $S$ is bounded and continuous, and
$b_{i,S}^\varepsilon(u_i^\varepsilon)$ converges to $S(u_i)$ a.e. in
$Q$ and in $L^\infty(Q)$ weak $\star$, ${\partial
b_{i,S}^\varepsilon(u_i^\varepsilon) \over \partial t}$ converges to
${\partial b_{i,S}(u_i) \over \partial t}$ in $D'(Q)$ as
$\varepsilon$ tends to $0$.
\textbf{Limit of $-\mathop{\rm
div}\Big(S'(u_i^\varepsilon)a_\varepsilon
(x,u_i^\varepsilon,Du_i^\varepsilon)\Big)$.} Since $\mathop{\rm
supp} S'\subset [-K,K]$, for $\varepsilon